Solutions to Ricci flow whose scalar curvature is bounded in L^p (II)

标量曲率以 L^p 为界的 Ricci 流的解 (II)

• 批准号: 339362328
• 负责人: Professor Miles Simon, Ph.D.
• 金额: --
• 依托单位: Institut für Analysis und Numerik (IAN)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/339362328?language=en
• 关键词: Solutions; Ricci; flow; whose; scalar

We plan to investigate the structure of finite time singularities of the Ricci flow in n dimensions if the scalar curvature remains globally or locally bounded in L^p for p>= n/2. In a preprint resulting from the first funding period of the project, the applicant and the Post-Doc Dr. Jiawei Liu, generalised results of earlier work of the applicant. We show in this preprint that if the solution is defined on a closed Kähler 2n-dimensional ( complex n-dimensional ) manifold, or a real four dimensional closed manifold, then:a) the integral L^q norm of of the Ricci curvature in space respectively L^s norm in space-time is bounded where q and s are explicit constants depending on n and p. b) the L^2 norm of the Riemannian curvature is bounded.We further show that the estimates of a) and b) can be localised in the Kähler setting. In four dimensions, we plan to use these integral estimates and concentration compactness methods of earlier work of the applicant, and where appropriate those of Bamler and Bamler-Zhang, to show that there is a local (or global in the real case) orbifold limit as t --> T, T being the singular time.In the Kähler local setting we plan to develop further these methods in general dimensions, and prove that there is a local metric space limit as t-->T, T being the singular time, and to study the structure thereof.We plan to develop a local orbifold Ricci flow in four dimensions and a local conical Ricci flow or other local weak Ricci flow in higher dimensions in order to continue the flow past the singular time.

我们计划研究n维Ricci流的有限时间奇点结构，如果标量曲率在L^p中保持全局或局部有界，p>= n/2。在项目第一个资助期的预印本中，申请人和博士后刘佳伟博士总结了申请人早期工作的成果。我们在这篇预印本中表明，如果解定义在一个封闭的Kähler 2n维（复n维）流形上，或者一个实的四维封闭流形上，那么：a)空间中里奇曲率的积分L^q范数在时空中的L^s范数是有界的，其中q和s是依赖于n和p的显式常数b)黎曼曲率的L^2范数是有界的。我们进一步表明，a)和b)的估计可以在Kähler设置中定位。在四个维度中，我们计划使用申请人早期工作的这些积分估计和集中紧致方法，以及适当的Bamler和Bamler- zhang的方法，来证明存在局部（或在实际情况中是全局）轨道极限t -> t， t是奇异时间。在Kähler局部环境下，我们计划在一般维度上进一步发展这些方法，并证明存在局部度量空间极限t- > t， t为奇异时间，并研究其结构。我们计划在四维空间发展局部的折弯Ricci流，在高维空间发展局部的锥形Ricci流或其他局部弱Ricci流，以使其在奇异时间后继续流动。